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PartialGeometry·1694

Kissing Number Problem

Ask how many equal spheres can touch one equal sphere without overlap.

Formula

Kissing number
τ(n)=max⁡{m:  ∃ u1,…,um∈Sn−1,  ⟨ui,uj⟩≤12  ∀ i≠j}\tau(n)=\max\{m:\;\exists\,u_1,\dots,u_m\in S^{n-1},\;\langle u_i,u_j\rangle\le\tfrac{1}{2}\;\forall\,i\ne j\}

The maximum number of non-overlapping unit spheres that can simultaneously touch one central unit sphere in Rⁿ.

Known exact values
τ(1)=2,τ(2)=6,τ(3)=12,τ(4)=24,τ(8)=240,τ(24)=196 560\tau(1)=2,\quad\tau(2)=6,\quad\tau(3)=12,\quad\tau(4)=24,\quad\tau(8)=240,\quad\tau(24)=196\,560

Exact values are known only in dimensions 1–4, 8, and 24. The τ(3) = 12 case settled Newton vs. Gregory (1694).

Delsarte LP bound
τ(n)≤max⁡{∑ifi:f≥0,  f^(k)≤0  (k≥1),  f(cos⁡θ)≤0  (θ<60°)}\tau(n)\le\max\Bigl\{\sum_i f_i: f\ge 0,\;\hat{f}(k)\le 0\;(k\ge 1),\;f(\cos\theta)\le 0\;(\theta<60°)\Bigr\}

Linear programming bounds on spherical codes — tight in dimensions 8 and 24, yielding the exact kissing numbers.

Summary

The kissing number problem is solved in several dimensions, including the famous 3D answer of 12, but remains open in many dimensions. It is a compact visual gateway into high-dimensional geometry.

Sources

The kissing number in four dimensions

arXiv:math/03094302003

Das Problem der dreizehn Kugeln

1953

The kissing number in four dimensions

2008

Videos

Thumbnail for Kissing Numbers - Numberphile

Kissing Numbers - Numberphile

Numberphile